Influence of neutron surface on E 1 resonance properties

The E1 strength distributions in even-even Si isotopes were calculated in the “particle-core coupling” version of the shell model taking into account the fragmentation of the hole configuration among the states of the daughter nuclei. The comparison of calculated strength distributions in different isotopes of the same element shows the peculiarities of a neutron surface influence on the E1 resonance fragmentation. The problem of theoretical description of the excited states of light nuclei has been discussed for many years. Various model approaches have achieved success in the study of highly excited nuclear states, primarily E1 resonance, as this state is the most studied experimentally [1]. From comparison of the experimental photo-excitation cross sections it is seen that a consistent increase in the number of neutrons in the isotopic chain changes drastically the resonance structure. Especially, these changes are critical in the area of light nuclei A < 50, where the resonance structure is important. Theoretical description of dipole resonances in some even isotopes of Ti and Ca was performed in the previous paper [2]. In this work the structure of excitations of 28Si and 30Si isotopes is discussed. 1 Particle-core coupling shell model The increasing amount of experimental data on structure of multipole giant resonances (GR) in the cross sections of nuclear reactions has shown that microscopic approach considering only particle-hole configurations cannot reproduce a complicated structure of GR. One of the possible ways to build a set of basic configurations which could be used as doorway states in the microscopic description of nuclear resonances is to take into account the distributions of the “hole” configurations among the states with total momentum J, excitation energy E and isospin T (JET )A−1 of residual nuclei with mass number (A − 1). In the Particle Core Coupling version of Shell Model (PCC SM) these distributions are taken into account in microscopic description of multipole resonances (MR) [3, 4]. Theoretical description of MR in 1p-shell nuclei based on the PCC SM has shown good agreement with experimental data for nuclei 7 ≤ A ≤ 15 [4]. The wave functions of excited states of a nucleus in the PCC SM constructed as a product of the wave functions of the final nucleus (A − 1) and the wave functions of the ae-mail: n.g.goncharova@gmail.com nucleon ∣∣∣J f T f 〉 = ∑ J′E′T ′, j f α J′E′T ′, j f f ∣∣∣(J′E′T ′)A−1 × ( j f ) : J f T f 〉 . (1) The set of states of the final nucleus in (1) has to include all states with the significant genealogical connection to the ground state of the target nucleus


Particle-core coupling shell model
The increasing amount of experimental data on structure of multipole giant resonances (GR) in the cross sections of nuclear reactions has shown that microscopic approach considering only particle-hole configurations cannot reproduce a complicated structure of GR.One of the possible ways to build a set of basic configurations which could be used as doorway states in the microscopic description of nuclear resonances is to take into account the distributions of the "hole" configurations among the states with total momentum J, excitation energy E and isospin T (JET ) A−1 of residual nuclei with mass number (A − 1).In the Particle Core Coupling version of Shell Model (PCC SM) these distributions are taken into account in microscopic description of multipole resonances (MR) [3,4].Theoretical description of MR in 1p-shell nuclei based on the PCC SM has shown good agreement with experimental data for nuclei 7 ≤ A ≤ 15 [4].
The wave functions of excited states of a nucleus in the PCC SM constructed as a product of the wave functions of the final nucleus (A − 1) and the wave functions of the a e-mail: n.g.goncharova@gmail.comnucleon The set of states of the final nucleus in (1) has to include all states with the significant genealogical connection to the ground state of the target nucleus The way to determine the probabilities of the various core states which appear when one of the nucleons would be extracted from the parent nucleus is to use the experimental data on the spectroscopic factors S i of direct pick-up reactions to estimate the coefficients of fractional parentage MGR probabilities could be calculated via the E1 form factors where the excitation matrix element is expressed in terms of matrix elements of one-nucleon transitions In Eq. ( 4), α f are the results of the Hamiltonian diagonalization.

Rigidities of Si isotopes
The distribution of hole states in the nucleus is highly influenced by the shape of nuclear potential well and its deviations from spheroid.In [5] was shown that the addition of neutron pairs to nucleus with given Z sometimes radically changes collective characteristics of nucleus.E.g. rigidity and hence surface tension of 48 Ca nucleus is about 10 times larger than these characteristics for 42 Ca or 44 Ca.
The larger surface tension results in larger compression on nuclear matter and such nuclei are closer to spherical shape.The experimental distributions of hole state strengths along the energy axis for such nuclei are close to shell-model predictions for spherical nuclei.On the other hand, in nuclei with a low surface tension the Coulomb force deforms the potential well.Comparison of rigidities C for the 28 Si and 30 Si isotopes is shown in Fig. 1 together with parameters r 0 of their charge radii R ch The value of rigidity for 30 Si is about 2 times larger than for 28 Si.The values of deformability for even-even Si isotopes are at least 10 times smaller then rigidity of 48 Ca.This difference in rigidities and, consequently, in surface tensions leads to great diversities in the distributions of the hole state strength.In Table 1, the occupancies N nl j = nl j S i in the 2s − 1d shell for 28 Si and 30 Si isotopes are shown.The values of N nl j were obtained from direct pick-up reaction data (corresponding evaluated data are taken from Refs.[6,7]).The (p, d) reactions on 28 Si show that not only 1d 5/2 but the 2s and 1d 3/2 hole states as well play an important role in excitation of this nucleus.For the  [8].This assumption agrees with the experimental value of quadrupole deformation β 2 = −0.42± 0.02 (see Ref. [9]).Comparison of the spectroscopic data for 30 Si with the Nilsson model predictions also suggests the negative deformation with β = −0.2[10].

PCC SM results for 28 Si and 30 Si
The results of PCC SM calculations for E1 in the isotopes 28 Si and 30 Si are shown in Fig. 2 and 3 together with the experimental data from [12].Figures 2(a   tions for 28 Si and 30 Si (thin lines) and experimental data [12] are presented.As seen from comparison of theory and experiment, calculations for both the nuclei reproduce the main peculiarities of E1 in the energy region below 22 − 23 MeV, i.e. at main peak and below.The "pygmy"resonance region in 28 Si is populated by transitions from 1d 3/2 and 2s subshells.The main peak of the E1 resonance in 28 Si is formed by transitions from 1d 5/2 to 1 f 7/2 state.The pronounced splitting of the main peak is a consequence of fragmentation of 1d 5/2 hole state.The emergence of an extra pair of neutrons in 30 Si leads to a considerable complication of the main E1 resonance peak structure due to additional transitions from the 2s and 1d 3/2 states.In the structure of the E1 "pygmy" resonance of 30 Si (at E < 18 MeV) the transition from 1d 3/2 hole dominates.
The region of E1 strength distribution above 24 − 25 MeV cannot be properly reproduced since deep hole states in A−1 nuclei were not revealed in pick up reactions performed with low projectile energies (see [6,7] and reference therein).
Excitation of E1 resonance in 30 Si leads to the appearance of states with the isospin values T > = 2 and T < = 1 (Fig. 3).For 30 Si the contributions of isospin branches to the cross section are comparable.The estimation for the difference between the average energies of T > and T < branches is about 7 MeV, which agrees with the assessment of isospin splitting based on experimental data [12].
The calculations of E1 strength distributions in 28 Si and 30 Si nuclei based on the Particle Core Coupling version of Shell Model show that the deviations of both nuclei from spherical form reveal in the splitting of main E1 peak and in population of "pygmy"-resonance area.Using experimental data on direct pick-up reaction spectroscopy allows one to trace the relationship of structural features of E1 resonance with effect of an extra pair of neutrons contribution.
) and 3(a) display the distributions of E1 form factors in 28 Si and 30 Si, respectively.Different types of shading indicate the type of transition giving a major contribution to F 2 (E exc ).In Figs.2(b) and 3(b), the calculated photo-neutron cross sec-

Figure 2 .
Figure 2. Form factors (a) and the estimated (γ, n) cross section (b) for 28 Si.Black squares -experimental cross sections from [12], solid line -the PCC SM calculations.

Figure 3 .
Figure 3. Form factors (a) and the estimated (γ, n) cross section (b) for30 Si.Black squares -experimental cross sections from[12]; the PCC SM calculations: the dashed line -T < contribution, the dash-dot line -the T > one, the solid line -the total result.